Doing some sleuthing and I can't say for sure because I don't have a clear primary source line but I suspect that your William W. Mansfield is listed as son of Mary A(nn) Mansfield (born 1818) living in Wayne Twshp, Jefferson County Ohio with sister Mary E and Thomas W in the 1850 and 1860 census. There are at least 3 other William Mansfields in Ohio of similar age. One is pretty well documented and not yours.
I make ths match as the "best guess" based on knowing that the 116th OVI and 117th OVI were raised about the same time and that the 116th had significant numbers of Belmont and Monroe County recruits and the 117th was raised primarily from Jefferson and Jackson Counties even though they are not neighbors to each other. Couple that with your William W's first born girl was named Mary A. it seems a logical fit and a place to start further hunting. I'd start there and dig around in the Jefferson County records to see if there is a will or probate record between 1840 and 1850 naming Mary A Mansfield. That could get you to a fathers name. As an aside that portion of Ohio was surveyed and numerous land grants were issued to Revolutionary and 1812 vets who pioneered the north side of the Ohio river in the 1810s to 1830s.